points M And M 1 are called symmetric with respect to a given line L if this line is the perpendicular bisector of the segment MM 1 (Figure 1). Each point of the line L symmetrical to itself. Plane transformation in which each point is mapped to a point symmetrical to it with respect to a given line L, is called axially symmetrical with the L axis and denoted S L :S L (M) = M 1 .

points M And M 1 are mutually symmetrical with respect to L, That's why S L (M 1 )=M. Therefore, the transformation inverse of axial symmetry is the same axial symmetry: S L -1= S L , S L° S L =E. In other words, the axial symmetry of a plane is involutive transformation.

The image of a given point with axial symmetry can be simply constructed using only one compass. Let L- axis of symmetry, A And B- arbitrary points of this axis (Fig. 2). If S L (M) = M 1 , then by the property of the points of the perpendicular bisector to the segment we have: AM=AM 1 And BM=BM 1 . So the point M 1 belongs to two circles: circles with center A radius AM and circles with center B radius BM (M- given point). Figure F and her image F 1 with axial symmetry are called symmetrical figures with respect to a straight line L(Figure 3).

Theorem. The axial symmetry of a plane is movement.

If A And IN- any points of the plane and S L (A)=A 1 , S L (B)=B 1 , then we have to prove that A 1 B 1 = AB. To do this, we introduce a rectangular coordinate system OXY so that the axis OX coincides with the axis of symmetry. points A And IN have coordinates A(x 1 ,-y 1 ) And B(x 1 ,-y 2 ) .Points A 1 and IN 1 have coordinates A 1 (x 1 ,y 1 ) And B 1 (x 1 ,y 2 ) (Figure 4 - 8). Using the formula for the distance between two points, we find:

From these relations it is clear that AB=A 1 IN 1 , which was to be proved.

From a comparison of the orientations of the triangle and its image, we obtain that the axial symmetry of the plane is movement of the second kind.

Axial symmetry maps each line to a line. In particular, each of the lines perpendicular to the axis of symmetry is mapped by this symmetry onto itself.

Theorem. A straight line other than a perpendicular to the axis of symmetry and its image under this symmetry intersect on the axis of symmetry or are parallel to it.

Proof. Let a straight line not perpendicular to the axis be given L symmetry. If m? L=P And S L (m)=m 1 , then m 1 ?m And S L (P)=P, That's why Pm1(Figure 9). If m || L, That m 1 || L, since otherwise the direct m And m 1 would intersect at a point on the line L, which contradicts the condition m||L(Figure 10).

By virtue of the definition of equal figures, straight lines, symmetrical about a straight line L, form with a straight line L equal angles (Figure 9).

Straight L called the axis of symmetry of the figure F, if with symmetry with the axis L figure F displayed on itself: S L (F)=F. They say that the figure F symmetrical about a straight line L.

For example, any straight line containing the center of a circle is the axis of symmetry of this circle. Indeed, let M- arbitrary point of the circle sch centered ABOUT, OL, S L (M)=M 1 . Then S L (O)=O And OM 1 =OM, i.e. M 1 є u. So, the image of any point of a circle belongs to this circle. Hence, S L (u)=u.

The axes of symmetry of a pair of non-parallel lines are two perpendicular lines containing the bisectors of the angles between these lines. The axis of symmetry of a segment is the line containing it, as well as the perpendicular bisector to this segment.

Axial symmetry properties

• 1. With axial symmetry, the image of a straight line is a straight line, the image of parallel lines are parallel lines
• 3. Axial symmetry preserves the simple ratio of three points.
• 3. With axial symmetry, the segment passes into a segment, a ray into a ray, a half-plane into a half-plane.
• 4. With axial symmetry, the angle goes into an equal angle.
• 5. With axial symmetry with the d-axis, any straight line perpendicular to the d-axis remains in place.
• 6. With axial symmetry, the orthonormal frame goes over into the orthonormal frame. In this case, the point M with coordinates x and y relative to the frame R goes to the point M` with the same coordinates x and y, but relative to the frame R`.
• 7. The axial symmetry of the plane translates the right orthonormal frame into the left one and, conversely, the left orthonormal frame into the right one.
• 8. The composition of two axial symmetries of a plane with parallel axes is a parallel translation by a vector perpendicular to the given lines, the length of which is twice the distance between the given lines

I . Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, withmeasures)

Summary table (all properties, features)

II . Symmetry Applications:

1) in mathematics

2) in chemistry

3) in biology, botany and zoology

4) in art, literature and architecture

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1. Basic concepts of symmetry and its types.

The concept of symmetry n R runs throughout the history of mankind. It is found already at the origins of human knowledge. It arose in connection with the study of a living organism, namely man. And it was used by sculptors as early as the 5th century BC. e. The word "symmetry" is Greek, it means "proportionality, proportionality, the sameness in the arrangement of parts." It is widely used by all areas of modern science without exception. Many great people thought about this pattern. For example, L. N. Tolstoy said: “Standing in front of a black board and drawing different figures on it with chalk, I was suddenly struck by the thought: why is symmetry clear to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on?" The symmetry is really pleasing to the eye. Who has not admired the symmetry of nature's creations: leaves, flowers, birds, animals; or human creations: buildings, technology, - all that surrounds us from childhood, that strives for beauty and harmony. Hermann Weyl said: "Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection." Hermann Weyl is a German mathematician. Its activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what signs to see the presence or, conversely, the absence of symmetry in a particular case. Thus, a mathematically rigorous representation was formed relatively recently - at the beginning of the 20th century. It is rather complicated. We will turn and once again recall the definitions that are given to us in the textbook.

2. Axial symmetry.

2.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to the line a if this line passes through the midpoint of the segment AA 1 and is perpendicular to it. Each point of the line a is considered symmetrical to itself.

Definition. The figure is said to be symmetrical with respect to a straight line. A, if for each point of the figure the point symmetrical to it with respect to the straight line A also belongs to this figure. Straight A called the axis of symmetry of the figure. The figure is also said to have axial symmetry.

2.2 Construction plan

And so, to build a symmetrical figure relative to a straight line from each point, we draw a perpendicular to this straight line and extend it by the same distance, mark the resulting point. We do this with each point, we get the symmetrical vertices of the new figure. Then we connect them in series and get a symmetrical figure of this relative axis.

2.3 Examples of figures with axial symmetry.

3. Central symmetry

3.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to the point O if O is the midpoint of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is called symmetric with respect to the point O if for each point of the figure the point symmetric to it with respect to the point O also belongs to this figure.

3.2 Construction plan

Construction of a triangle symmetrical to the given one with respect to the center O.

To construct a point symmetrical to a point A relative to the point ABOUT, it suffices to draw a straight line OA(Fig. 46 ) and on the other side of the point ABOUT set aside a segment equal to a segment OA. In other words , points A and ; In and ; C and are symmetrical with respect to some point O. In fig. 46 built a triangle symmetrical to a triangle ABC relative to the point ABOUT. These triangles are equal.

Construction of symmetrical points about the center.

In the figure, the points M and M 1, N and N 1 are symmetrical about the point O, and the points P and Q are not symmetrical about this point.

In general, figures that are symmetrical about some point are equal to .

3.3 Examples

Let us give examples of figures with central symmetry. The simplest figures with central symmetry are the circle and the parallelogram.

Point O is called the center of symmetry of the figure. In such cases, the figure has central symmetry. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals.

The line also has central symmetry, however, unlike the circle and the parallelogram, which have only one center of symmetry (point O in the figure), the line has an infinite number of them - any point on the line is its center of symmetry.

The figures show an angle symmetrical about the vertex, a segment symmetrical to another segment about the center A and a quadrilateral symmetrical about its vertex M.

An example of a figure that does not have a center of symmetry is a triangle.

4. Summary of the lesson

Let's summarize the knowledge gained. Today in the lesson we got acquainted with two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.

Summary table

	Axial symmetry	Central symmetry
Peculiarity	All points of the figure must be symmetrical with respect to some straight line.	All points of the figure must be symmetrical about the point chosen as the center of symmetry.
Properties	1. Symmetric points lie on perpendiculars to the line. 3. Straight lines turn into straight lines, angles into equal angles. 4. The sizes and shapes of the figures are saved.	1. Symmetrical points lie on a straight line passing through the center and the given point of the figure. 2. The distance from a point to a straight line is equal to the distance from a straight line to a symmetrical point. 3. The sizes and shapes of the figures are saved.

II. Application of symmetry

© Sukhacheva Elena Vladimirovna, 2008-2009

You will need

• - properties of symmetrical points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compass;
• - pencil;
• - paper;
• - a computer with a graphics editor.

Instruction

Draw a line a, which will be the axis of symmetry. If its coordinates are not given, draw it arbitrarily. On one side of this line, put an arbitrary point A. you need to find a symmetrical point.

Helpful advice

Symmetry properties are constantly used in the AutoCAD program. For this, the Mirror option is used. To build an isosceles triangle or an isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Mirror them with the specified command and extend the sides to the required size. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, this will be a given value.

You constantly come across symmetry in graphic editors when you use the “flip vertically / horizontally” option. In this case, a straight line corresponding to one of the vertical or horizontal sides of the picture frame is taken as the axis of symmetry.

Sources:

• how to draw central symmetry

Constructing a section of a cone is not such a difficult task. The main thing is to follow a strict sequence of actions. Then this task will be easy to do and will not require much effort from you.

You will need

• - paper;
• - pen;
• - circle;
• - ruler.

Instruction

When answering this question, you first need to decide what parameters the section is set to.
Let this be the line of intersection of the plane l with the plane and the point O, which is the point of intersection with its section.

The construction is illustrated in Fig.1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. As a result, point L is obtained. Further, through point O, draw a straight line LW, and build two directing cones lying in the main section O2M and O2C. At the intersection of these guides lie the point Q, as well as the already shown point W. These are the first two points of the required section.

Now draw a perpendicular MC at the base of the cone BB1 ​​and build the generators of the perpendicular section O2B and O2B1. In this section, draw a straight line RG through t.O, parallel to BB1. T.R and t.G - two more points of the desired section. If the cross section of the ball were known, then it could be constructed already at this stage. However, this is not an ellipse at all, but something elliptical, having symmetry with respect to the segment QW. Therefore, you should build as many points of the section as possible in order to connect them in the future with a smooth curve to get the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and build the corresponding guides O2A and O2N. Through PO draw a straight line passing through PQ and WG, until it intersects with the newly constructed guides at points P and E. These are two more points of the desired section. Continuing in the same way and further, you can arbitrarily desired points.

True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, it is possible to draw straight lines SS' parallel to RG in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It suffices to construct half of the required section due to the already mentioned symmetry with respect to QW.

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Tip 3: How to Graph a Trigonometric Function

You need to draw schedule trigonometric functions? Master the algorithm of actions using the example of building a sinusoid. To solve the problem, use the research method.

You will need

• - ruler;
• - pencil;
• - Knowledge of the basics of trigonometry.

Instruction

Related videos

note

If the two semi-axes of a one-lane hyperboloid are equal, then the figure can be obtained by rotating a hyperbola with semi-axes, one of which is the above, and the other, which differs from two equal ones, around the imaginary axis.

Helpful advice

When considering this figure with respect to the axes Oxz and Oyz, it is clear that its main sections are hyperbolas. And when a given spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The throat ellipse of a one-strip hyperboloid passes through the origin, since z=0.

The throat ellipse is described by the equation x²/a² +y²/b²=1, and the other ellipses are composed by the equation x²/a² +y²/b²=1+h²/c².

Sources:

• Ellipsoids, paraboloids, hyperboloids. Rectilinear Generators

The shape of the five-pointed star has been widely used by man since ancient times. We consider its form to be beautiful, since we unconsciously distinguish the ratios of the golden section in it, i.e. the beauty of the five-pointed star is justified mathematically. Euclid was the first to describe the construction of a five-pointed star in his "Beginnings". Let's take a look at his experience.

You will need

• ruler;
• pencil;
• compass;
• protractor.

Instruction

The construction of a star is reduced to the construction and subsequent connection of its vertices to each other sequentially through one. In order to build the correct one, it is necessary to break the circle into five.
Construct an arbitrary circle using a compass. Mark its center with an O.

Mark point A and use a ruler to draw line segment OA. Now you need to divide the segment OA in half, for this, from point A, draw an arc with radius OA until it intersects with a circle at two points M and N. Construct a segment MN. Point E, where MN intersects OA, will bisect segment OA.

Restore the perpendicular OD to radius OA and connect point D and E. Make notch B on OA from point E with radius ED.

Now, using the segment DB, mark the circle into five equal parts. Mark the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the points in the following sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is the correct five-pointed star, into a regular pentagon. It was in this way that he built

Goals:

• educational:
  • give an idea of ​​\u200b\u200bsymmetry;
  • introduce the main types of symmetry in the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand ideas about famous figures by introducing them to the properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate the acquired knowledge;
• general education:
  • learn to set yourself up for work;
  • teach to control oneself and a neighbor on the desk;
  • to teach how to evaluate yourself and a neighbor on your desk;
• developing:
  • activate independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • educate students "a sense of shoulder";
  • cultivate communication;
  • inculcate a culture of communication.

DURING THE CLASSES

In front of each are scissors and a sheet of paper.

Exercise 1(3 min).

- Take a sheet of paper, fold it in half and cut out some figure. Now unfold the sheet and look at the fold line.

Question: What is the function of this line?

Suggested answer: This line divides the figure in half.

Question: How are all the points of the figure located on the two resulting halves?

Suggested answer: All points of the halves are at an equal distance from the fold line and at the same level.

- So, the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is the axis of symmetry.

Task 2 (2 minutes).

- Cut out a snowflake, find the axis of symmetry, characterize it.

Task 3 (5 minutes).

- Draw a circle in your notebook.

Question: Determine how the axis of symmetry passes?

Suggested answer: Differently.

Question: So how many axes of symmetry does a circle have?

Suggested answer: A lot of.

- That's right, the circle has many axes of symmetry. The same wonderful figure is the ball (spatial figure)

Question: What other figures have more than one axis of symmetry?

Suggested answer: Square, rectangle, isosceles and equilateral triangles.

– Consider three-dimensional figures: a cube, a pyramid, a cone, a cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry a square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute the halves of plasticine figures to the students.

Task 4 (3 min).

- Using the information received, finish the missing part of the figure.

Note: the figurine can be both flat and three-dimensional. It is important that students determine how the axis of symmetry goes and fill in the missing element. The correctness of the performance is determined by the neighbor on the desk, evaluates how well the work has been done.

A line is laid out from a lace of the same color on the desktop (closed, open, with self-crossing, without self-crossing).

Task 5 (group work 5 min).

- Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.

The correctness of the work performed is determined by the students themselves.

The students are presented with elements of drawings

Task 6 (2 minutes).

Find the symmetrical parts of these drawings.

To consolidate the material covered, I propose the following tasks, provided for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What are the types of these triangles?

2. Draw in a notebook several isosceles triangles with a common base equal to 6 cm.

3. Draw a segment AB. Construct a line perpendicular to segment AB and passing through its midpoint. Mark points C and D on it so that the quadrilateral ACBD is symmetrical with respect to the line AB.

- Our initial ideas about the form belong to a very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions that differed little from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and in the late Paleolithic era, they decorated their existence by creating works of art, figurines and drawings, which reveal a wonderful sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity enters a new stone age, the Neolithic.
Neolithic man had a keen sense of geometric form. The firing and coloring of clay vessels, the manufacture of reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
Where is symmetry found in nature?

Suggested answer: wings of butterflies, beetles, tree leaves…

“Symmetry can also be seen in architecture. When constructing buildings, builders clearly adhere to symmetry.

That's why the buildings are so beautiful. Also an example of symmetry is a person, animals.

Homework:

1. Come up with your own ornament, depict it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, mark where there are elements of symmetry.

Human life is filled with symmetry. It is convenient, beautiful, no need to invent new standards. But what is she really and is she as beautiful in nature as is commonly believed?

Symmetry

Since ancient times, people have sought to streamline the world around them. Therefore, something is considered beautiful, and something not so. From an aesthetic point of view, golden and silver sections are considered attractive, as well as, of course, symmetry. This term is of Greek origin and literally means "proportion". Of course, we are talking not only about coincidence on this basis, but also on some others. In a general sense, symmetry is such a property of an object when, as a result of certain formations, the result is equal to the original data. It is found in both animate and inanimate nature, as well as in objects made by man.

First of all, the term "symmetry" is used in geometry, but finds application in many scientific fields, and its meaning remains generally unchanged. This phenomenon is quite common and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in ornaments on fabric, building borders and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely exciting.

Use of the term in other scientific fields

In the future, symmetry will be considered from the point of view of geometry, but it is worth mentioning that this word is used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon is studied from different angles and under different conditions. The classification, for example, depends on which science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged everywhere.

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Classification

There are several basic types of symmetry, of which three are most common:

In addition, the following types are also distinguished in geometry, they are much less common, but no less curious:

• sliding;
• rotational;
• point;
• progressive;
• screw;
• fractal;
• etc.

In biology, all species are called somewhat differently, although in fact they can be the same. The division into certain groups occurs on the basis of the presence or absence, as well as the number of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

Some features are distinguished in the phenomenon, one of which is necessarily present. The so-called basic elements include planes, centers and axes of symmetry. It is in accordance with their presence, absence and quantity that the type is determined.

The center of symmetry is called the point inside the figure or crystal, at which the lines converge, connecting in pairs all sides parallel to each other. Of course, it doesn't always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since there is none. According to the definition, it is obvious that the center of symmetry is that through which the figure can be reflected to itself. An example is, for example, a circle and a point in its middle. This element is usually referred to as C.

The plane of symmetry, of course, is imaginary, but it is she who divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or it can divide them. For the same figure, several planes can exist at once. These elements are usually referred to as P.

But perhaps the most common is what is called "axes of symmetry." This frequent phenomenon can be seen both in geometry and in nature. And it deserves separate consideration.

axes

Often the element with respect to which the figure can be called symmetrical,

is a straight line or a segment. In any case, we are not talking about a point or a plane. Then the axes of symmetry of the figures are considered. There can be a lot of them, and they can be located in any way: divide sides or be parallel to them, as well as cross corners or not. Axes of symmetry are usually denoted as L.

Examples are isosceles and equilateral triangles. In the first case, there will be a vertical axis of symmetry, on both sides of which there are equal faces, and in the second, the lines will intersect each corner and coincide with all bisectors, medians, and heights. Ordinary triangles do not have it.

By the way, the totality of all the above elements in crystallography and stereometry is called the degree of symmetry. This indicator depends on the number of axes, planes and centers.

Examples in Geometry

It is conditionally possible to divide the entire set of objects of study of mathematicians into figures that have an axis of symmetry, and those that do not. All regular polygons, circles, ovals, as well as some special cases automatically fall into the first category, while the rest fall into the second group.

As in the case when it was said about the axis of symmetry of the triangle, this element for the quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram, it is, but for an irregular figure, accordingly, it is not. For a circle, the axis of symmetry is the set of straight lines that pass through its center.

In addition, it is interesting to consider volumetric figures from this point of view. At least one axis of symmetry, in addition to all regular polygons and the ball, will have some cones, as well as pyramids, parallelograms and some others. Each case must be considered separately.

Examples in nature

Mirror symmetry in life is called bilateral, it is most common
often. Any person and very many animals are an example of this. The axial one is called radial and is much less common, as a rule, in the plant world. And yet they are. For example, it is worth considering how many axes of symmetry a star has, and does it have them at all? Of course, we are talking about marine life, and not about the subject of study of astronomers. And the correct answer would be this: it depends on the number of rays of the star, for example, five, if it is five-pointed.

In addition, many flowers have radial symmetry: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.

Arrhythmia

This term, first of all, reminds most of medicine and cardiology, but it initially has a slightly different meaning. In this case, the synonym will be "asymmetry", that is, the absence or violation of regularity in one form or another. It can be found as an accident, and sometimes it can be a beautiful device, for example, in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous Leaning Tower of Pisa is slightly tilted, and although it is not the only one, this is the most famous example. It is known that this happened by accident, but this has its own charm.

In addition, it is obvious that the faces and bodies of people and animals are also not completely symmetrical. There have even been studies, according to the results of which the "correct" faces were regarded as inanimate or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore extremely interesting.